A Bayesian Approach to inferring vascular tree structure from 2D imagery
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We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo (MCMC) algorithm is employed to produce approximate samples from the posterior distribution given local feature estimates, derived from likelihood
ABAYESIANAPPROACHTOINFERRINGVASCULARTREESTRUCTUREFROM
2DIMAGERY
ElkeTh¨onnes,AbhirBhalerao,WilfridKendall,andRolandWilson
DepartmentsofComputerScienceandStatistics
UniversityofWarwick,UK
elke|wsk@stats.warwick.ac.uk
abhir|rgw@dcs.warwick.ac.uk
ABSTRACT
Wedescribeamethodforinferringtree-likevascularstruc-turesfrom2Dimagery.AMarkovChainMonteCarlo(MCMC)algorithmisemployedtoproduceapproximatesamplesfromtheposteriordistributiongivenlocalfeatureestimates,derivedfromlikelihoodmaximisationforaGaus-sianintensitypro le.Amultiresolutionscheme,inwhichcoarsescaleestimatesareusedtoinitialisethealgorithmfor nerscales,hasbeenimplementedandusedtomodelreti-nalimages.Resultsarepresentedtoshowtheeffectivenessofthemethod.
1.INTRODUCTION
Theproblemofinferringvascularstructurefromimagedataisanimportantone,especiallyintheareaofsurgicalplan-ning,whichrequiresbothef cientcomputationandeffec-tiveuseofpriorknowledge.Previousworkintheareahastendedtofocusonthemodellingofspeci cvascularfea-tures[1]ortouseapproachessuchasadaptivethresholding[4].
Theaimoftheworkdescribedhereistoformulateageneralmethodfortheinference,whichcanbeappliedintwoorthreedimensionsandmakeseffectiveuseofpriorknowledge,yetwhichissuf cientlygeneraltobeappliedtoawiderangeofproblems.Thecommonstatisticalmeth-odsforsuchmedicalimageanalysishavetypicallyusedlikelihoodtechniques,suchasExpectation-Maximisation(EM)[6,5].AlthoughEMmethodscanbeef cientcom-putationally,theyhaveonlylimitedscopeforincorporatingpriorknowledge.AmorepowerfulwayofincludingpriorinformationistouseaBayesianmethod,suchasmaximumaposteriori(MAP)estimation.Theprincipaldif cultywithBayesiantechniquesisacomputationalone:theynormallyrequiretheuseofMarkovchainMonteCarlo(MCMC)al-gorithms,whichmayrunforhundredsofthousandsofiter-ationstoyieldreliableresults[3].Thishasrestrictedtheir
We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo (MCMC) algorithm is employed to produce approximate samples from the posterior distribution given local feature estimates, derived from likelihood
displacementtobeasmalllinearmultipleofthedisplace-mentoftheparentvertexfromthegrandparentvertex(an“AR(1)”tree)(righthandof gure1).Tomodelintensity,witheachvertexinthetreeweassociateaGaussianker-nelthatrepresentsthespatialgreylevelpro leofthecorre-spondingvesselsegment.
ofTheposteriordistributionforarandomnumber
treesisgivenby
(1)
whereistheimage.Thedistributionof,,penalisesthenumberoftrees;intheexamples,aPoissondistributionwasused.Thisensuresthata‘minimal’expla-nationofthedataisfound.aretheprobabilitiesde ningthedegreeofthebranchingprocessandisthedegreeofthevertex.isthedistribu-tionoftheparametersofthevertex,.Intheexamples,thisisanautoregressive(AR(1))process.Theparametersrepresentthepositionsofvertices,,andtheampli-tudeandwidthparametersoftheedges,whileisthelikelihoodfunction.TheobservationmodelisbasedontheapproximationoflinearstructuresbyasumofGaussianker-nels:
parent
We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo (MCMC) algorithm is employed to produce approximate samples from the posterior distribution given local feature estimates, derived from likelihood
Fig.2.MovesusedinMCMCsimulationontrees.iterative,EM-typealgorithmtomaximiselikelihood
(4)
(5)
(6)
wherethedataarewindowedwithacosinewindow,whosesizeistwicetheblockwidthatagivenscale,togivethedatausedintheestimator.Theindexdenotesiterationnumber;typically4-5iterationsaresuf- cienttogiveaccurateestimates.Figures3(b)-(c)showreconstructionsusingthe2DGaussiansineachblock(atcorrespondingblocksizes)basedontheMLfeatureesti-matesatblocksizesofandrespectively.Clearly,atlowerspatialresolutions,themodelcannoteas-ilydescribethepresenceofmultiplevesselswithinthewin-dow,suchasoccuratbifurcations,andtheresultinglow-amplitude,isotropicGaussiansarelocallythe‘best’de-scriptionoftheseregions.However,theseblockscanbemodelledaccuratelyathigherspatialresolutions.Thesec-ondsetofimagesshowshowthelocalestimatesfromdif-ferentwindowscalesisusedinacoarse- nestochasticsim-ulation,inordertogetaBayesianestimateoftheforest
structure.Afterthe rst200iterations,thescaleishalvedandtheappropriatelocalfeatureestimatesareusedtoguidethesampler;after3000,thescaleishalvedagain,asitisafter6000iterations,atwhichpoint,thehighestspatialres-olutionisreached.Thisapproachhasbeenfoundtospeedconvergencetotheequilibriumdistribution,whileavoidingbecomingtrappedinlocalmodes,inasimilarmannertomanycoarse- nealgorithms.Notethatoneiterationcon-sistsofthegenerationandacceptanceofasingleproposal(for‘editing’thetree).Inotherwords,100000iterationsiscomparable,intermsofcomputation,toasinglescanthroughtheimage.Ithasbeennotedfromexperimentsthatequilibriumisreachedinapproximately50000iterations,acomparativelylowburdencomputationally.
4.CONCLUSIONS
Someencouragingpreliminaryresultshavebeenachievedusingtheapproachdescribedinsection2,demonstratingitspotentialformodellingvascularstructuregloballyinacomputationallyef cientway.Fine-tuningthealgorithmwillleadtosigni cantimprovements.Thesewillinclude,forexample,theuseofthelocalestimatestoproduceinitialcon gurationsfortheMCMCalgorithm.Suchimprove-mentsarecurrentlybeingimplemented.Theworkisalsobeingtestedonothertypesofdataandextendedtothreedimensions.
5.REFERENCES
[1]P.DatlingerA.Pinz,S.BernoggerandA.Kruger.Map-pingthehumanretina.IEEETrans.MedicalImaging,17(1):606–619,1998.[2]S.P.Brooks.TheMarkovChainMonteCarloMethod
anditsApplication.TheStatistician,47:69–100,1998.[3]W.R.Gilks,S.Richardson,andD.J.Spiegelhalter.
MarkovChainMonteCarloinPractice.Chapman&Hall,1996.[4]M.E.Martinez-Perez,A.D.Hughes,A.V.Stanton,
A.S.Thom,A.A.Bharath,andK.H.Parker.Seg-mentationofretinalbloodvesselsbasedontheseconddirectionalderivativeandregiongrowing.InProc.ofIEEEICIP-99,pages173–176,Kobe,Japan,1999.[5]W.M.Wells,R.Kikinis,W.E.L.Grimson,and
R.Jolesz.AdaptivesegmentationofMRIdata.IEEETrans.MedicalImaging,15:429–442,1996.[6]D.L.WilsonandJ.A.Noble.Anadaptivesegmen-tationalgorithmforextractingarteriesandaneurysmsfromtime-of- ightMRAdata.IEEETrans.MedicalImaging,18(10):938–945,1999.
We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo (MCMC) algorithm is employed to produce approximate samples from the posterior distribution given local feature estimates, derived from likelihood
(a)
(b)
(c)
Fig.3.(a)2Dretinalangiogramsize404404pixels.(b)Reconstructionofdatafrommodelparametersestimates
forblocksizesof64and(c)
16.
(a)
(b)
(c)
Fig.4.Estimatesfromthetree-basedsampler,at(a)200,(b)1000and(c)50000iterations,showinghowuseismadeofthelocalmultiresolutionfeatureestimates,inacoarse- neapproachtotheMCMCalgorithm.After50000iterations,fewchangesoccur.
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